For brainiest:):):):):):):):)

2) Use spherical coordinates to find the integral of f(x,y,z) = z over x^2 + y^2 + z^2 < 4 and x,y,z < 0

777dan777 [17]

1. Denote by $\mathcal D$ the region bounded by the cylinder $x^2+y^2=9$ and the planes $z=0$ and $z=6$ . Converting to cylindrical coordinates involves setting

$\begin{cases}x=r\cos u\\y=r\sin u\\z=v\end{cases}$

and $\mathcal D$ is obtained by varying the parameters over $0\le r\le3$ , $0\le u\le2\pi$ , and $0\le v\le6$ . The volume element is

$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm du\,\mathrm dv$

so the integral becomes

$\displaystyle\iiint_{\mathcal D}xz\,\mathrm dV=\int_{v=0}^{v=6}\int_{u=0}^{u=2\pi}\int_{r=0}^{r=3}r^2v\cos u\,\mathrm dr\,\mathrm du\,\mathrm dv=0$

2. Now let $\mathcal D$ denote the region bounded by the sphere $x^2+y^2+z^2=4$ and the coordinate planes in the octant in which each coordinate of the point $(x,y,z)$ is negative.

Converting to spherical coordinates, we set

$\begin{cases}x=\rho\cos u\sin v\\y=\rho\sin u\sin v\\z=\rho\cos v\end{cases}$

where we obtain $\mathcal D$ by varying over $0\le\rho\le2$ , $\pi\le u\le\dfrac{3\pi}2$ , and $\dfrac\pi2\le v\le\pi$ . The volume element is now

$\mathrm dV=\rho^2\sin v\,\mathrm d\rho\,\mathrm du\,\mathrm dv$

so the integral becomes

$\displaystyle\iiint_{\mathcal D}z\,\mathrm dV=\int_{v=\pi/2}^{v=\pi}\int_{u=\pi}^{u=3\pi/2}\int_{\rho=0}^{\rho=2}\rho^3\cos v\sin v\,\mathrm d\rho\,\mathrm du\,\mathrm dv=-\pi$

8 0

3 years ago

What is x^4-4x^3-32 in factored form

marishachu [46]

Factoring: x4-4x3+8x-32

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: 8x-32
Group 2: -4x3+x4

Pull out from each group separately :

Group 1: (x-4) • (8)
Group 2: (x-4) • (x3)
-------------------
Add up the two groups :
(x-4) • (x3+8)

3 0

2 years ago

QUESTION SHOWN ON PHOTO :)

worty [1.4K]

Answer:

A: -1/3x

Step-by-step explanation:

We can convert -1/3 to -2/6 by multiplying -1/3 by 2/2. As we know, 5-2 is 3. That makes 3/6. 3/6 is equivalent to 1/2. That means it has to be A.

6 0

2 years ago

Every that is fully readable

n200080 [17]

10)
60in=60in × (1ft/12in) = 5ft
30in= 30in × (1ft/12in) = 2.5ft
12in = 12in × (1ft/12in) = 1ft
answer is = 5ft × 2.5ft× 1ft =12.5 ft^3 = a

11)
dozen = 12 hammers
12 hammers = 20 pounds
therefore 1 hammer= 20/12 = 1.6666 lb

21 hammers ×1.666 = 35 = c

12)
convert qt to gal
600 gal × 4 qt/gal = 2400 qt

2400 qt ÷ 120 qt/min
=20 min = d

13)
Carla is 12
rob is 12+5= 17
cousin is 12-7 = 5

12+17+5= 34 =a

15)
convert 60ft into yards
60 ft × 0.333 yd/ft =20 yd

diameter = 20 yd

area of circle = (pi/4)×d^2
(pi/4)×20^2= 314.16 = b

8 0

3 years ago

Area addition and subtraction

telo118 [61]

Answer:Area of the shaded region is 73.6 cm^2

Step-by-step explanation:

The circle is divided into two sectors. The Smaller sector contains the triangle. The angle that the smaller sector subtends at the center of the circle is 80 degrees. Since the total angle at the center of the circle is 360 degrees, it means that the angle that the larger sector subtends at the center would be 360 - 80 = 280 degrees

Area of a sector is expressed as

Area of sector = #/360 × πr^2

# = 280

r = 5 cm

Area of sector = 280/360 × 3.14 × 5^2

Area of sector = 61.06 cm^2

Area of the triangle is expressed as

1/2bh = 1/2 × 5 × 5 = 12.5

Area of the shaded region = 61.06 +

12.5 = 73.6

7 0

3 years ago